The incomplete beta function plays an important role in statistics in connection with the beta distribution (see, for instance, [1]}). Large parameter asymptotic approximations are useful in these applications. For fixed \(x\)  and \(b\) , one could use the asymptotic expansion
\(I_x(a,b)=\frac{x^a(1-x)^{b-1}}{a B(a,b)}\,{}_2F_1\left(\begin{matrix} {1,1 - b} \\ {a + 1} \end{matrix};\frac{x}{x-1}\right)\sim \frac{x^a(1-x)^{b-1}}{a B(a,b)} \sum _{n=0}^\infty \frac{\left(1-b\right)_n}{\left(a+1\right)_n}\left(\frac{x}{x-1}\right)^n,\) 
